Julià Cufí

The index of a plane curve and green’s formula

AbstractIn this paper we compute the line integral of a complex function on a rectifiable cycle homologous to zero obtaining a Green’s formula with multiplicities that involves the

A note on Hurwitz's inequality☆

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Evolutes and isoperimetric deficit in two-dimensional spaces of constant curvature

of the function and the index of the cycle. We consider this formula in several settings and we obtain a sharp version in terms of the Lebesgue integrability properties of the partial derivatives of the function. This result depends on the proven fact that the index of a rectifiable cycle is square integrable with respect to the planar Lebesgue measure.

Cauchy kernels in strictly pseudoconvex domains and an application to a mergelyan type approximation problem

In this paper we provide a Bonnesen-style inequality which gives a lower bound for the isoperimetric deficit corresponding to a closed convex curve in terms of some geometrical invariants of this curve. Moreover we give a geometrical interpretation for the case when equality holds.

Characterizing Abelian Admissible Groups

Abstract Given a simple closed plane curve Γ of length L enclosing a compact convex set K of area F , Hurwitz found an upper bound for the isoperimetric deficit, namely L 2 − 4 π F ≤ π | F e | , where F e is the algebraic area enclosed by the evolute of Γ. In this note we improve this inequality finding strictly positive lower bounds for the deficit π | F e | − Δ , where Δ = L 2 − 4 π F . These bounds involve either the visual angle of Γ or the pedal curve associated to K with respect to the Steiner point of K or the L 2 distance between K and the Steiner disk of K . For compact convex sets of constant width Hurwitzs inequality can be improved to L 2 − 4 π F ≤ 4 9 π | F e | . In this case we also get strictly positive lower bounds for the deficit 4 9 π | F e | − Δ . For each established inequality we study when equality holds. This occurs for those compact convex sets being bounded by a curve parallel to an hypocycloid of 3, 4 or 5 cusps or the Minkowski sum of this kind of sets.

The calculation of the L2-norm of the index of a plane curve and related formulas

We prove a characterization of some

A general form of Green Formula and Cauchy Integral Theorem

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